Sony is one-fifth the age of her mother 15 years ago and Sony’s brother is three-fifth the age of his mother 10 years ago. If the sum of Sony and her brother’s ages is 31 then how old is Sony's mother ?

Options

A. A) 40
B. B) 50
C. C) 60
D. D) 70

Answer

To find out how old Sony's mother is, we can break down the problem step by step.

### Step 1: Define Variables
Let \( x \) be the current age of Sony's mother.

### Step 2: Determine Sony's Age
According to the problem, Sony is one-fifth the age of her mother 15 years ago. Therefore, we can express Sony's age as:
\[ \text{Sony's age} = \frac{1}{5} \times (x - 15) \]

### Step 3: Determine Brother's Age
Similarly, Sony's brother is three-fifth the age of his mother 10 years ago. Thus, we can express his age as:
\[ \text{Brother's age} = \frac{3}{5} \times (x - 10) \]

### Step 4: Set Up the Age Equation
We know that the sum of Sony's and her brother's ages is 31:
\[ \frac{1}{5} \times (x - 15) + \frac{3}{5} \times (x - 10) = 31 \]

### Step 5: Simplify the Equation
To simplify the equation, we can combine the fractions:
\[ \frac{1}{5}(x - 15) + \frac{3}{5}(x - 10) = \frac{1}{5}(x - 15 + 3(x - 10)) \]
\[ = \frac{1}{5}(x - 15 + 3x - 30) = \frac{1}{5}(4x - 45) \]
So, we have:
\[ \frac{4x - 45}{5} = 31 \]

### Step 6: Clear the Fraction
Multiply both sides by 5 to eliminate the fraction:
\[ 4x - 45 = 155 \]

### Step 7: Solve for \( x \)
Now, add 45 to both sides:
\[ 4x = 155 + 45 \]
\[ 4x = 200 \]
Now, divide by 4:
\[ x = 50 \]

### Conclusion
Thus, Sony's mother is **50 years old**.

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